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Figure 12.10 The energies of the lowest conduction-band state (left) and highest valenceband state (right) as functions of the superlattice period
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Following on from this, Fig. 12.10 displays the results of calculations of the energies at the centre of the superlattice Brillouin zone (k = 0) for the uppermost valence band and the lower-most conduction band, as a function of the superlattice period (expressed in units of A0). In both cases, the quantum-confinement energy decreases as the superlattice period increases, an effect which was commonly observed in the discussions in s 2 and 3. Note that the energies are given in relation to the top of the valence band of bulk GaAs, which is set to zero, with the bottom of the conduction band coming out as 1.499 eV in these calculations. One interesting feature is the sudden reduction in the quantum-confinement energy of the conduction-band state when the period decreases to just A0. In fact, it would be expected a priori that the conduction band minimum would tend towards the band edge of the bulk alloy Ga 0.5 Al 0.5 As in the limit of short-period GaAs/AlAs superlattices. By taking a linear combination of the bulk band structures as suggested by the virtual crystal approximation, this would imply an energy of around 2 eV. This behaviour will be discussed in more detail in the next section. Finally for this series of calculations, Fig. 12.11 plots the charge density along the centre (x = y = 0) of the nz=10 superlattice spiral, for the uppermost valenceband state. The confinement in the GaAs layers is clear, and in addition to this, the lowest conduction-band state is also confined in these layers, thus illustrating the Type-I nature of the band alignment. It can be seen from this figure that the wave function does consist of two components, where one is rapidly varying and the second is an envelope. This reflects well the idea behind the envelope-function approximation which was discussed (and then used extensively) earlier. The rapidly varying component has a period which is the same as the atomic spacing, and indeed along the axis used in the plot, i.e. the x = y = 0 axis, the peaks in the charge density correspond to bond centres. The envelope function is obtained by joining together the peaks in the charge-density plot by using the eye, with its form being familiar from 2.
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Figure 12.11 The charge density of the uppermost valence-band state along the x = y = 0 (z-) axis for two complete superlattice periods
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12.3 COMPARISON WITH ENVELOPE-FUNCTION APPROXIMATION Fig. 12.12 compares the results of a Kronig-Penney superlattice calculation under the envelope function/effective mass approximations (see Section 2.13), with the empirical pseudopotential calculations of the lowest-energy conduction band state as a function of the superlattice period, as previously displayed in Fig. 12.10. It is this comparison which draws attention to the complexity of the data. For guidance, it is necessary to refer to the specialist treatise on empirical pseudopotential calculations of GaAs/AlAs superlattices from which the pseudopotentials employed in these calculations are taken (see [253], p. 17396).
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Figure 12.12 Comparison of the lowest-energy conduction band state given by the empirical pseudopotential (EPP) calculation in the previous section with that given by the Kronig-Penney model of a superlattice under the envelope function/effective mass approximations (EF/EMA)
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Mader and Zunger indicate that these simple bulk-like potentials are not sufficient to describe the microscopic structure of such a short-period superlattice and hence the result for nz=1 could be prone to error. In particular, no account has been made for the change in the coordination of the interface anions (see Section 12.5). For 2 < nz < 8, the conduction-band minimum for the superlattice originates from the X valleys of the bulk and not from the F valley this is possible because of the indirect nature of AlAs. As the X valley in bulk AlAs lies below that of the F valley, then narrow quantum wells can produce such high confinement energies that the eigenstate is influenced by these outlying valleys (in essence!). For nz > 8, the simpler Kronig-Penney model describes the behaviour of the energy state with superlattice period quite well qualitatively.
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Figure 12.13 Comparison of the highest-energy valence-band state given by the empirical pseudopotential (EPP) calculation in the previous section with that given by the Kronig-Penney model of a superlattice under the envelope function/effective mass approximations (EF/EMA)
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Fig. 12.13 displays the results of an identical series of calculations for the uppermost (heavy-hole) valence-band state. The effective mass in the AlAs barriers was taken (and fixed) as the bulk value of 0.51 m0 (Adachi [14], p. 254). In this case, however, the hole effective mass in the GaAs well regions was used as a parameter and varied in order to produce the best fit to the empirical pseudopotential (EPP) results. It can be seen that the Kronig-Penney (EF/EMA) results for an effective mass of 0.45m0 fit the pseudopotential data very well at the larger well and barrier widths, but the match is poorer at narrower widths. This is the result found by Long et al. [21], and indeed should be expected, as the envelope-function approximation (see 1) hinges on the point that the wave function can be considered as a product of two components, with one being a rapidly varying Bloch function (which is factorised out) and the second a slowly varying envelope. It stands to reason that when the period of the envelope approaches that of the Bloch function, as happens here in short-period superlattices, the approximation becomes poorer.
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